| L(s) = 1 | + 8·2-s + 18·3-s + 48·4-s + 53·5-s + 144·6-s + 256·8-s + 243·9-s + 424·10-s + 191·11-s + 864·12-s + 379·13-s + 954·15-s + 1.28e3·16-s + 340·17-s + 1.94e3·18-s + 1.76e3·19-s + 2.54e3·20-s + 1.52e3·22-s + 3.23e3·23-s + 4.60e3·24-s − 1.74e3·25-s + 3.03e3·26-s + 2.91e3·27-s + 4.45e3·29-s + 7.63e3·30-s − 1.99e3·31-s + 6.14e3·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.948·5-s + 1.63·6-s + 1.41·8-s + 9-s + 1.34·10-s + 0.475·11-s + 1.73·12-s + 0.621·13-s + 1.09·15-s + 5/4·16-s + 0.285·17-s + 1.41·18-s + 1.12·19-s + 1.42·20-s + 0.673·22-s + 1.27·23-s + 1.63·24-s − 0.557·25-s + 0.879·26-s + 0.769·27-s + 0.984·29-s + 1.54·30-s − 0.372·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(23.25512880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(23.25512880\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 53 T + 4552 T^{2} - 53 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 191 T + 24656 p T^{2} - 191 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 379 T + 488066 T^{2} - 379 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 20 p T + 1908514 T^{2} - 20 p^{6} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1769 T + 2083758 T^{2} - 1769 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3236 T + 15452206 T^{2} - 3236 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4459 T + 37061638 T^{2} - 4459 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 1994 T - 6304813 T^{2} + 1994 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 20587 T + 238401006 T^{2} - 20587 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8814 T + 240678562 T^{2} + 8814 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 15853 T + 338678796 T^{2} - 15853 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 33912 T + 687783466 T^{2} + 33912 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 49239 T + 1320998110 T^{2} - 49239 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 56735 T + 2230528834 T^{2} - 56735 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 67508 T + 2826067262 T^{2} + 67508 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 75723 T + 3861149404 T^{2} - 75723 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8992 T - 681216182 T^{2} + 8992 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3201 T + 2311013380 T^{2} - 3201 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 26612 T + 3997648985 T^{2} - 26612 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 949 T + 367057696 T^{2} - 949 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 176562 T + 18855802834 T^{2} + 176562 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 129423 T + 11942811256 T^{2} - 129423 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.18079705849768582115916838910, −10.90188447971699712849803674165, −10.06692101871903669819079946885, −9.807158247010878719365031544446, −9.373660577129236932390163105811, −8.845121445594351049477797087457, −8.123435997814502019991591874434, −7.86212993282366863671818867780, −6.98735982932894800674244015423, −6.84524716143414639851243455596, −6.08156794920723170357358317201, −5.66837638939159672045978117534, −5.09481266461612846990856491548, −4.48838434061096796824316261062, −3.79479341123492627653582509376, −3.44863478789883915223006110258, −2.55526013655289765776987636997, −2.48038006936622756390708073092, −1.32330637735012584549227107731, −1.10968052726208394639219179181,
1.10968052726208394639219179181, 1.32330637735012584549227107731, 2.48038006936622756390708073092, 2.55526013655289765776987636997, 3.44863478789883915223006110258, 3.79479341123492627653582509376, 4.48838434061096796824316261062, 5.09481266461612846990856491548, 5.66837638939159672045978117534, 6.08156794920723170357358317201, 6.84524716143414639851243455596, 6.98735982932894800674244015423, 7.86212993282366863671818867780, 8.123435997814502019991591874434, 8.845121445594351049477797087457, 9.373660577129236932390163105811, 9.807158247010878719365031544446, 10.06692101871903669819079946885, 10.90188447971699712849803674165, 11.18079705849768582115916838910
